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Cut The Knot!
by Alex Bogomolny
Impressions from ICME-9
At the beginning of the month I had the gratifying experience of attending ICME-9. ICME - the International Congress on Mathematics Education - convenes every four years in a different country. This year the Congress was held in Makuhari, Japan. An interesting fact: people who attended all congresses starting with the first one are known as old hands. Three old hands were in attendance at ICME-9: J. Becker (USA), C. Gaulin (Canada), and E. Wittman (Germany). For me it was the first time in Japan and at ICME. My thanks go to NSF and NCTM for the travel grant that made this trip possible.
Makuhari is located some 35 km from Tokyo's main subway hub. How far is this? (I just remembered the musings of Sei Shonagon - a court lady in 10th century Japan - about things distant and near.) It depends, of course. By train it's about a 30 minute ride, but a highway drive may take more than 2 hours. Traffic is very heavy there.
And another reflection on the distance. The weather there was very hot and humid. The sun was scorching even with dark and heavy clouds making their way across the sky. Twice during the congress, the CNN forecasts promised rain in Tokyo and environs. And indeed we were close enough to hear the rattle of thunder and catch glimpses of lightning, but rains never reached Makuhari. (Not until the last day of the congress when the weather finally relented did a few drops reach the ground.)
With all the reservations and registrations completed over the Internet months in advance, getting a registration bundle was a breeze, the weather notwithstanding. A few thoughts crossed my mind on browsing an information sheet. The Congress drew 1840 mathematics educators from 75 countries. The former Soviet Union was represented by four now independent republics: Russia (11 participants), Ukraine (5), Latvia (2), and Estonia (1). 7 participants from former Czechoslovakia were evenly distributed between the Czech Republic (4) and Slovakia (3). Three participants came from Poland - formerly of the Warsaw Pact, now a NATO member. Another 3 came from Slovenia - a republic in Yugoslavia until a few years ago. Here are some additional statistics:
|Number of participants
|Population (in thousands)
(There were three separate delegations: the mainland China (111), China - Hong Kong (21), China - Taiwan (17).)
Many saw significance in that the Congress was held in Asia for the first time. While mathematics education evolves in every country on a national background, research is truly international. Emphasis on problem solving skills seems to be a common denominator for various schools of mathematics education. In Japan, The Three Open Methods will be included into the national high school curriculum starting with 2002. The three are open process, open-end product, open problem formulation. The first captures the intention to pursue multiple ways of solving the same problem. The second refers to the problems with more than one right answer. The third encourages students to pose and solve their own problems.
Aside from 4 plenary and 45 partially overlapping regular lectures, the work at the congress was organized into 13 Working Groups for Action and 23 Topic Study Groups. TSGs ran in a more or less standard format: lectures followed by question-answer sessions. WGAs split their allotted time between presentations and small group discussions. I took part in WGA1 (Mathematics Education in Pre- and Primary School) and TSG7 (The Use of Multimedia in Mathematics Education.)
According to Mogens Niss (Denmark) who gave the first plenary lecture Key Issues and Trends in Research on Mathematics Education, the field of mathematics education research has reached its first stage of maturity. This is especially manifest in the breadth and scope of current research. In the 1960s and 1970s, the first researchers were preoccupied with mathematical curriculum and ways of teaching it. Next, the goals and objectives of mathematics education became objects of discussion and investigation. Various teaching aids and information technologies fell into the research domain as soon as they were available. Investigation of curriculum development process grew out of consideration for societal and institutional matters. Issues related to the education and profession of mathematics teachers were included into research agendas since the 1970s (education) and 1980s (profession). Research into students' actual learning processes began gaining momentum in the 1980s. In the 1990s, students' notions and beliefs with respect to mathematics were added to research programs. Today, research on the learning of mathematics is probably the predominant type of research in the field of mathematics education. Current research also covers classroom communication, social, cultural, and linguistic influences on the teaching and learning of mathematics, and of course its assessment. According to Niss,
In the 70s and 80s it became essential to researchers in mathematics education to make sure that no factor which might exert significant influence on the teaching and learning of mathematics were excluded from theoretical and empirical consideration. This was often formulated as the necessity of avoiding unjustified reduction of complexity.
While this is always of crucial importance in any field of research, including ours, we should not forget that the ultimate goal of any scientific endeavour is to achieve justified reduction of complexity. ...
One aspect of this is that we have focused on understanding what happens in and to individual or few students, as a necessary prerequisite to any future work.
In elementary education there is an overwhelming concern with psychological aspects of child development. The prevalent research trend embraces Jean Piaget's constructivism and Lev Vygotsky's social constructivism. The former refers to Piaget's doctrine that knowledge cannot be transmitted to an individual but must be constructed by the individual through personal exploration. The latter emphasizes the crucial role of conversation and, more generally, social interaction between individual learners.
Both with regard to instruction and assessment, the logistics of a constructivist educational system is anything but simple. Judging from the WGA1 discussions, teachers are more comfortable with a solid curriculum that offers well defined goals. The reasons are many. In the context of instruction, letting students construct their own knowledge at their own pace makes taxing demands on teachers' mastery of the subject matter. As was noted by Margaret Brown (King's College, UK),
(Recent studies) ... are demonstrating that teachers whose children have the lowest learning gains are those who use transmission teaching which aims at standard techniques, and those who use discovery styles of teaching which emphasise readiness and manipulatives. The most successful teachers have a commitment to making connections, both between different mathematical ideas and children's current state of understanding.
However, making connections on individual levels of understanding calls for flexibility of content knowledge that comes with a reasonable proficiency in the subject matter. The latter is usually not assumed of elementary school teachers who instruct in several subjects.
In the context of assessment, individual treatment of students in one-on-one informal, unstructured sessions is on one hand prohibitively time consuming, while on the other necessitates competence in aspects of child psychology traditionally not required of elementary school teachers.
Lastly, education is a multistage process. Every stage assumes a certain degree of knowledge presumably acquired in previous stages. A well defined curriculum plays the role of a goal setter for admission to and successful performance in the next stage. Unless the whole system undergoes a fundamental overhaul, we may expect that teachers will be more comfortable with curriculum as a framework for specific content and target skills than as a set of open-ended activities.
One observation that a mathematics educator can hardly avoid to make is the widening gap between researchers and practitioners in mathematical education. The very existence of a gap is neither surprising nor worrying. The course for concern is that it is widening.
If we are unsuccessful in this, research on mathematical education runs the risk of becoming barren dry swimming, while the practice of teaching runs the risk of becoming more naïve, narrow-minded, and inefficient than necessary and desirable.
In another plenary lecture, Erich Wittman (Germany) introduced an idea of substantial learning environments (SLE) as a tool for bridging the gap between theory and practice in mathematics education. The lecture was preceded by a video segment showing two girls probably 6-7 years of age playing the game of Scoring.
Wittman defines SLE as a teaching/learning unit with the following properties:
- It represents central objectives, contents and principles of teaching.
- It is related to significant mathematical contents, processes and procedures, and is a rich source for mathematical activities.
- It is flexible and can be adapted to the conditions of special classrooms.
- It involves mathematical, psychological and pedagogical aspects of teaching mathematics in a holistic way, and offers a wide potential for empirical research.
Scoring can be used to introduce and practice counting (forward and backwards), counting by grouping, division with remainder, modular arithmetic, binary system, logical and bitwise operations, impartial games, and more. The right strategy and winning and losing positions may be empirically discovered by students in early grades.
Wittman warned however of a danger inherent in focusing mathematics education on SLEs:
Substantial mathematics is fundamentally related to mathematical processes such as mathematising, exploring, reasoning and communicating. Emphasising these higher skills can easily lead to neglecting basic skills. Basic skills also tend to be neglected for another reason: In their eagerness to get rid of stereotyped forms of teaching and learning in favor of "constructivist" forms reformers easily get trapped. They tend to identify practice with stereotyped practice and by abolishing stereotypes they do away with practice of skills at all.
Seymour Papert [The Children's Machine, BasicBooks, 1993, p 1] offers a thought experiment. Imagine two groups (surgeons and teachers) of time travellers from 100 years ago brought into our time. Surgeons are shown to an operating room, teachers to a classroom. Who would be more surprised by the observable changes in profession? His answer: the surgeons would probably not know what their colleagues were trying to accomplish with those unfamiliar devices. The teachers might be puzzled by the changes. They might disagree whether the changes were for the better or for the worse, but other than that they would be right at home in a modern classroom.
There are many reasons why that would be so. Instead of discussing them, I only want to remark that, as rumors have it, there always have been and there still are good teachers out there whose students grow to love and appreciate mathematics even without becoming mathematicians by trade. These good teachers achieve their results in a more or less traditional classroom framework. I believe that the field of mathematics education could greatly benefit from documenting and classifying experiences and practices that reliably achieve very desirable results. If the field of mathematics education is to become a science in its next stage of maturity, the future science will be classified as natural rather than speculative. I strongly believe that, as a side effect, concentration on gathering and classification of past and current experiences may well help close the gap between research and practice in mathematics education.
In his book (pp 151-153), Seymour Papert gives a short Description of Piaget's three developmental stages (sensorimotor, concrete operations, formal thinking). He notes that Piaget failed to recognize that concrete thinking and knowing is not confined to children in a certain age group:
Children do it, people in Pacific and African villages do it, and so do the most sophisticated people in Paris and Geneva.
At the closing of WGA1, participants were asked to suggest topics for discussion at ICME-10. As I am sure that teachers also do and benefit from, concrete thinking, I suggested a discussion on good teaching practices. Hopefully, somebody will respond to this idea.
Copyright © 1996-2018 Alexander Bogomolny